Physics Today Von Klitzing Wins Nobel Physics Prize for Quantum Hall EffectBertram Schwarzschild Citation: Physics Today 38(12), 17 (1985); doi: 10.1063/1.2814805 View online: http://dx.doi.org/10.1063/1.2814805 View Table of Contents:http://scitation.aip.org/content/aip/magazine/physicstoday/38/12?ver=pdfcov Published by the AIP Publishing

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IIon Mining wins Nobel Physics Prize for quantum Hall effectie Royal Swedish Academy of Sci-lces has awarded the 1985 Nobel

in Physics to Klaus von Klitzing,, director of the Max Planck Institutejr Solid-State Research in Stuttgart,ie 1.8-million-kroner ($230 000) prize

ras awarded for his discovery of thejuantized Hall effect in 1980. Von

litzing, who was at the time a Heisen-3rg fellow at the University of Wiirz-

burg, discovered the effect at the high-Id magnet laboratory in Grenoble, a

lint facility of the Max Planck Insti-lte and the CNRS.The quantized Hall effect, a remar-able macroscopic quantum pheno-lenon occurring at low temperature inra-dimensional electron systems sub-ected to high magnetic fields, has inie past five years initiated consider-

experimental and theoretical ac-|ivity. Its most striking manifestation

the appearance of Hall-conductivityiteaus at integral multiples of e2/h

nth an astonishing precision quitelanticipated by the theorists—and

[juite oblivious to the imperfections orDmetric details of the semiconductor

iterfaces at which the effect is mea-lred (PHYSICS TODAY, June 1981, page

17).Von Klitzing's accidental observa-

;ion of conductivity plateaus whileiring semiconductor carrier den-

sities and the elementary theory ofindau-level filling suggested to himit one might exploit the Hall effect to

chieve a high-precision measuremente2/h, a fundamental constant of

iture, and hence ultimately a bettersrmination of the fine-structure

Dnstant, independent of the electron's^magnetic ratio. Not only wouldprovide pure physics with a strin-

ent test of quantum electrodynamics,; would also serve the applied-physics

engineering communities by fur-ling a fundamental, easily repro-

iucible standard of electrical resis-lce. To the nearest ohm, hie2 has alue of 25 813 ft. The International

[Bureau of Weights and Measures is[Considering a redefinition of the stan-fdard ohm in terms of the quantum Hall

feet.'In the theory, e2/h is the fundamen-

[ tal unit of [two-dimensional] conductiv-ity," von Klitzing told us, "but nobody

VON KLITZING

recognized that you can measure itdirectly from the Hall effect—probablybecause theorists don't know that onecan measure the Hall conductivityindependently of the geometry of thesystem. My idea was to make sample-independent measurements at filledLandau levels."

Even if the theorists had thoughtabout how to do such a measurement,they would not have anticipated theexistence of broad conductivity pla-teaus yielding (by now) e2/hto almost apart in ten million. After the firstquantum-Hall-effect results were pub-lished1 five years ago by von Klitzingand his colleagues Gerhardt Dorda(Siemens) and Michael Pepper (Cam-bridge), the scramble was on to find atheoretical explanation for why mea-surements at a crystal interface full ofimperfections yield such astonishinglyprecise agreement with a naive theorythat lightly waves these imperfectionsaside.

Von Klitzing saw the first indica-tions of the Hall-conductivity plateausas he was measuring carrier densitiesin 1978. At first he thought this anuninteresting, sample-dependent ef-fect. But when he noticed that theplateaus were always occurring at thesame values, he began to make theconnection to e2/h. When he moved hisexperiment from Grenoble to Wiirz-

burg to investigate the precision of theeffect with a carefully calibrated resis-tance standard, von Klitzing told us, heassumed the plateaus would agree withne2/h only within a few percent. Heforesaw nothing like the part-per-mil-lion precision reported in the 1980paper.

Having thoroughly digested thequantized-Hall-effect data, the theo-rists, have reached something of aconsensus that, far from being a dis-turbing complication, the localizedelectron states due to lattice imperfec-tions are essential to the observation ofthe effect. In a perfect crystal inter-face, they tell us, one would see no Hall-conductivity plateaus at all. From the1974 theory of Tsuneya Ando (Univer-sity of Tokyo), which largely ignoredlocalization, one would expect the Hallconductivity to measure the electrondensities of filled Landau levels."But," points out Livermore theoristRobert Laughlin, "with lots of electronsimmobilized by lattice imperfections,you'd never get that kind of precisionwith a quantized density measurement.In any case, the broad plateaus tell usthat whatever we're measuring is unaf-fected by adding lots of electrons. I'mnow convinced that what you're mea-suring in the quantized Hall effect isthe charge of the electron itself."

The experimenters have allowed thetheorists little respite. In the spring of1982, Daniel Tsui, Horst Stormer andArthur Gossard at Bell Labs, havingalready replicated2 von Klitzing's re-sult in a somewhat different setting,discovered3 something quite new—thefractional quantum Hall effect (PHYSICSTODAY, July 1983, page 19). This exten-sion of the effect to fractional multiplesof e2fh "knocked our socks off," remem-bers Laughlin.

The fractional quantum Hall effectpresented a much subtler theoreticalpuzzle than did von Klitzing's integralquantum Hall effect, as we must nowcall it. "Theoretical progress has beenmuch slower here," Laughlin told us,"and nightmarishly difficult experi-ments are still needed to confirm thetheory many of us think is the rightone." Just as in the case of the integralHall effect, Laughlin's theory asserts,one is measuring the charge of the

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carrier—in this case the fractionallycharged excitations of a novel quantumliquid4 (see page 89).

The classical Hall effect is essentiallythe tendency of charged particles incrossed magnetic and electric fields todrift sideways, that is, in the directionorthogonal to both fields. If, for exam-ple, one imposes a perpendicular mag-netic field on a current-carrying con-ducting strip, the Lorentz force willtend to pile up the moving electronsnear one edge of the strip, producing atransverse "Hall voltage" across itssurface. In the extreme, dissipation-free case of noninteracting electrons ina vacuum, the only net motion is in theE x B direction. The electrons will ingeneral execute cyclotron orbitsaround the magnetic-field lines withthe cyclotron frequency coc = eB/m,but the net motion is a cycloid sidewaysdrift at the "Hall velocity," E/B. Aver-aged over times large compared with 1/coc, the electric field does no net workon the electrons in the absence ofdissipative collisions.

In the quantized Hall effect, just sucha dissipation-free situation can occureven in a semiconductor crystal—whenall the conduction electrons find them-selves in fully occupied Landau levels.All the quantum-Hall experiments in-volve a two-dimensional system of elec-trons trapped at an interface—in vonKlitzing's case, the inversion layer inthe silicon just below the oxide surfaceof a metal-oxide-semiconductor field-effect transistor (MOSFET) at tempera-tures below 2 K. The low temperatureassures that all the electrons are in theground state of the trapping potentialwell at the silicon surface; they are freeto move only in the plane of theinversion layer. If one now imposes astrong magnetic field normal to theinversion layer, this ground statebreaks up into Landau levels as theelectrons execute tiny cyclotron orbitsin the plane. An electron in the nthLandau level will gain a cyclotron-orbite n e r g y of (n — 1/2>fieoc •

There is a strict upper limit on thedensity of electrons in any one Landaulevel. For a given magnetic fieldstrength imposed on such a two-dimen-sional sea of conduction electrons, aLandau level is fully occupied when thetwo-dimensional density is l/irrc

2,where rc = (2fi/eB)1/2, the radius of acyclotron orbit of energy ficoc. One canthink of this as the tightest possiblepacking of cyclotron orbits in theplane—or alternatively, one electronper magnetic-flux quantum per Lan-dau level.

For an arbitarily chosen populationdensity and magnetic field at low tem-perature, some highest Landau levelwill in general be partially filled, withall lower-lying Landau levels fully oc-cupied. If, in this general case, one

The first report of thequantized Hall effect,as published by vonKlitzing, Dorda and

Pepper in 1980. Asthe population of the

MOSFET inversion layerincreases with

increasing gate voltage(/?), the Hall

resistance, asmeasured by the

voltage (UH)transverse to the

current, falls byquantized, stepwise

plateaus. Thelongitudinal voltage(Um) falls almost to

zero at each plateau,indicating lossless

current flow when theLandau levels are full.

UH/mVp-SUBSTRATE

HALL PROBE

0;- n=0

•Vg /V

starts a current flowing in the inver-sion layer by imposing an electric fieldin the plane, one will see an unspecta-cular manifestation of the Hall effect.The current flow will have componentsboth parallel and perpendicular to theelectric field. The conductivity is gen-eralized to a two-by-two matrix. Its off-diagonal element axy, the "Hall con-ductivity," is the current density divid-ed by the electric-field componenttransverse to it. Although axy may betechnologically interesting as a mea-sure of the carrier density available inthe semiconducting sample under scru-tiny, its complicated dependence on thenature and geometry of the samplerenders it—in the general case—some-thing less than fundamental.

What von Klitzing demonstrated is thataxy does indeed take on a very funda-mental and sample-independent char-acter in the special case where thehighest occupied Landau level is full.In that case—assuming a perfect inter-face—the current flow must be com-pletely nondissipative. The only stateinto which a moving electron could bescattered would be the first unoccupiedLandau level. But that would involvethe electron's jumping an energy gap officoc, an essentially impossible event atthe low temperatures and high magnet-ic fields of these experiments. Thesituation is then like that of the vacu-um. The current flows only perpendic-ular to any electric field in the plane,drifting at the Hall velocity, E/B. Thecurrent density is then simply the

combined charge density of the n fullyoccupied Landau levels times the Hallvelocity. Dividing by the electric fieldto get the Hall conductivity, one can-cels all dependence on field strengthsand gets simply axy = ne2/h, indepen-dent of everything except the funda-mental constants. The question thatremains is: What does one see at a real,imperfect lattice interface, where theFermi level that determines the energysurfaces of the two-dimensional sea ofconduction electrons has local irregu-larities?

Measuring the Hall conductivity tohigh precision is relatively straightfor-ward. Von Klitzing's MOSFETS em-ployed oxide-covered high-mobility sili-con strips of various geometries sup-plied by Dorda and Pepper. Attemperatures below 2 K a fixed currentwas made to flow along the surfaceinversion layer of the silicon strip(typically a few hundred microns long)from source to drain, while probesmeasured the longitudinal voltage dropalong the current direction and thetransverse Hall voltage across thestrip. In the two-dimensional case withlossless current flow, axy is simply thereciprocal of the measured Hall resis-tance. In the Wiirzburg experiments,von Klitzing measured this Hall resis-tance very accurately by calibratingthe 10-kfl reference resistor in thedevice against a standard maintainedat the Physikalisch Technische Bund-esanstalt in Braunschweig. In this wayhe could measure the Hall conductivity

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to about a part in a million. Theoriginal discovery was made with a 20-tesla field at the Grenoble magnetlaboratory, but the 15-tesla magnetsavailable at Wiirzburg were sufficientfor the subsequent high-precision mea-surements.

The conductivity plateaus that charac-terize the quantum Hall effect appearin von Klitzing's experiment as one fillssuccessive Landau levels at a fixedmagnetic field by steadily raising thepopulation of conduction electrons inthe inversion layer. One accomplishesthis by increasing the MOSFET gatevoltage that provides the trapping elec-tric field normal to the interface, andthus continuously raising the popula-tion, and hence the Fermi level, of thetwo-dimensional electron sea. As theFermi level rises to fill a particularLandau level, the Hall conductivity, asplotted against the gate voltage, in-creases steadily and unspectacularly.But then, presumably when the «thLandau level is completely filled, theHall conductivity suddenly levels off toform a wide, spectacularly flat plateauat a value of ne2/h, accurate to about apart in ten million, until the nextLandau level begins filling.

Repeating this exercise with MOSFETSof various geometries, von Klitzingfound no observable difference fromone sample to the next. When hemonitored the longitudinal voltagedrop along the current, he found, asexpected from the naive theory, thatthe longitudinal resistance drops al-most to zero at each Hall-conductivityplateau; the current flow is essentiallylossless when all the occupied Landaulevels are full.

Shortly thereafter, Tsui and Gossardat Bell Labs achieved2 the same resultwith a modulation-doped GaAs/Al-GaAs heterojunction in place of vonKlitzing's silicon MOSFET. In this case,however, the mobile electron popula-tion at the interface is fixed by thedoping of the AlGaAs layer and thebending of the conduction band at theinterface. Therefore, instead of vary-ing a gate voltage, the Bell Labs groupvaried the filling of the Landau levelsby varying the imposed magnet field.As B increases for fixed population, thecyclotron orbits get tighter and themaximum allowed density of each Lan-dau layer becomes correspondinglylarger.

The GaAs heterostructure offers sev-eral advantages over silicon MOSFETSfor further quantum-Hall-effect re-search. Because the effective electronmass is significantly smaller in GaAs,the energy separation fuoc betweenLandau levels is correspondingly larg-er. Therefore one can see the effect atlower magnetic fields (8 T) and highertemperatures (4 K). Even more impor-tant, one can grow much cleaner inter-

faces in a GaAs heterostructure. Thefractional quantum Hall effect, discov-ered the following year by Tsui andStormer, has been seen at Bell Labsonly in the very cleanest heterostruc-tures grown by Gossard and JamesHwang.3 For this completely unantici-pated discovery, Tsui, Stormer andGossard were awarded the 1984 OliverBuckley Prize for Condensed MatterPhysics (PHYSICS TODAY, March 1984,page 107).

In the naive theory, the steplike Hallplateaus found by von Klitzing presenta puzzle. The electrons in each Landaulevel are delocalized along equipoten-tials of the Hall field. In the absence oflocalized electron states, all the elec-trons would be ensconced in Landaulevels, the Fermi level would jumpdirectly from one Landau level to thenext and the Hall conductivity wouldequal ne2/h only at singular points.Perhaps the plateaus arise because theFermi level becomes temporarilypinned between Landau levels by local-ized electron states that are due tolattice imperfections. But if dirt isdoing the trick, how can one explainthe extraordinary part-per-ten-millionaccuracy of the plateau conductivitylevels?

After von Klitzing's discovery, Uni-versity of Maryland theorist RichardPrange undertook5 to explain the unex-pected accuracy of the quantized Halleffect by treating lattice imperfectionsas strong delta-function scatterers in asimple, calculable model. He conclud-ed that whereas localized impuritystates broken off from the Landaulevels carry no current, the extendedelectron states in the Landau levelscompensate precisely for this loss ofcarriers by carrying just enough extracurrent to preserve the precision of thequantized Hall effect.

His calculation was, however, highlymodel dependent, unrealistically neg-lecting electron-electron interactionsand Landau-level mixing. Arguingthat so general a phenomenon mustcome from general principles, not de-tails, Laughlin has produced6 a model-independent derivation of the quantumHall effect. Laughlin's argument,based essentially on gauge invariance,seems now to express something of aconsensus among theorists working inthis area. Considering a Gedankenex-periment in which a two-dimensionalHall current flows around a strip thatis closed upon itself to form a loop andthreaded by a magnetic flux, he askswhat happens when that flux is adiaba-tically increased, one flux quantum at atime. Extended electron states thatpersist coherently around the loop willcontribute to the current induced bythe increasing flux. But because thesestates are pinned in phase by closingupon themselves, they can respond to

the increasing flux only by transferringan integral number of electrons perflux quantum from one edge of the stripto the other. The resulting transverseHall voltage gives precisely von Klitz-ing's result. The Hall conductivity isessentially measuring the charge of theelectron. Localized states, on the otherhand, can have arbitrary phase undergauge transformation and thus do notcontribute to the Hall current. "Add-ing electrons after a Landau level isfilled does nothing," Laughlin explains."They're stuck in localized states.Without localization by lattice imper-fections, you'd see no effect. The quan-tum Hall effect is in fact the mostimportant consequence of localizationthat's ever been discovered."

Serge Luryi and Rudolf Kazarinov atBell Labs have produced7 a heuristical-ly appealing percolation-model deriva-tion in which the distinction betweenextended and localized electron statesis purely topological. Both states areregarded as wavefunction fibers ex-tending along equipotentials, but it isthe topologically "global" states thattake on the entire applied voltage. Asthe Fermi level rises gradually over apotential surface made uneven by lat-tice imperfections, the conductivityplateaus result from topological trans-formations of the landscape "fromlakes on a landmass to islands in a sea."A virtue of this model, Luryi told us, isthat it explains the plateaus withoutrequiring a high density of imperfec-tions and truly localized states.

None of the theoretical work on theintegral quantum Hall effect offeredthe slightest inkling or explanation ofthe fractional quantum Hall effect thatwas soon to follow.

Von Klitzing believes that "we have yetto find a complete microscopic theory ofthe integral quantum Hall effect." Atthe Stuttgart Max Planck Institute heis now working primarily on thin-layerand quantum-well phenomena, mostlyin GaAs heterostructures. Althoughhe himself is not involved in growingthese structures, his move last winterfrom the Technical University of Mu-nich was in part impelled by themolecular-beam-epitaxy facilities atStuttgart. He had joined the Munichphysics faculty shortly after his histor-ic 1980 experiment.

Von Klitzing received his PhD atWiirzburg in 1972. In 1981 the Ger-man Physical Society awarded him itsWalter Schottky Prize for Solid StateResearch for his discovery of the quan-tum Hall effect. "I am happy thatsemiconductor physics has once againbeen recognized with a Nobel Prize," hetold us. "I hope that this will lead—atleast in Germany—to more support ofsolid-state physics in this area. InGermany, no company does the kind ofbasic research one sees at Bell Labs or

PHYSICS TODAY / DECEMBER 1985 19

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IBM; only the Max Planck Instituteand a few university groups." As forthe prospect that the Stuttgart areamight become a new Silicon Valley, hetold us, "they're trying, but there's nodecision as yet."

—BERTRAM SCHWARZSCHILD

References1. K. V. Klitzing, G. Dorda, M. Pepper,

Phys. Rev. Lett. 45, 494 (1980).2. D. C. Tsui, A. R. Gossard, Appl. Phys.

Lett. 38, 550 (1981).3. D. Tsui, H. Stormer, A. Gossard, Phys.

Rev. Lett. 48, 1559 (1982).

4. R. Laughlin, Phys. Rev. Lett. 50, 1395(1983).

5. R. Prange, Phys. Rev. B 23, 4802 (1981).6. R. Laughlin, Phys. Rev. B 23, 5652

(1981).7. S. Luryi, R. Kazarinov, Phys. Rev. B 27,

1386 (1983).

Nobel Prize in Chemistry to Hauptman and KarleAlthough the Nobel Prize has beenfrequently awarded to researchers whohave applied the techniques of x-raycrystallography to the determination ofmolecular structure, x-ray crystallog-raphy as a mature science has seldombeen recognized by the Nobel commit-tee. This oversight was redressed whenthe 1985 Nobel Prize in Chemistry wasawarded to Herbert A. Hauptman ofthe Medical Foundation of Buffalo,New York, and Jerome Karle of the USNaval Research Laboratory in Wash-ington, D.C., "for their outstandingachievements in the development ofdirect methods for the determination ofcrystal structure." Many chemists andbiologists who rely heavily on themethods developed by Hauptman andKarle feel that the prize was longoverdue.

Before the pioneering work1 ofHauptman and Karle in the early1950s, the determination of the three-dimensional structure of a moderatelylarge molecule proceeded by indirecttrickery. Molecules that did not con-tain heavy atoms took months or yearsto solve—if, indeed, they were solved atall. Crystals that did not have a centerof symmetry were generally ignored.But the direct method of Hauptmanand Karle, run on modern computers,allows the structures of mid-sized mol-ecules to be routinely determined inabout two days. "Nowadays the methodis used everywhere," says Hauptman,who estimates that between 40 000 and50 000 structures have been deter-mined using their method—with 5000new structures a year being added tothe list.

Karle's wife, Isabella, made a vitalcontribution to the further develop-ment of the direct methods invented byKarle and Hauptman by tackling thegeneral problem of determining thestructure of crystals that do not havecenters of symmetry. Her work led tothe so-called symbolic addition proce-dure, which was used to determine2 thefirst noncentrosymmetric crystal struc-ture, L-arginine dihydrate. The sym-bolic addition procedure has since beenapplied to molecules that contain asmany as 200 independent atoms. Jer-ome Karle says that "the major accom-plishments in the development andapplication of the symbolic additionprocedure are hers."

The phase problem. Diffraction of x

HAUPTMAN

rays is the most convincing demonstra-tion of the existence of the regulararrangement of atoms in a crystal. Thevariously oriented atomic planes in acrystal each reflect x rays towardunique spots on an x-ray diffractionphotograph. The pattern of diffractionspots can therefore be used to identifythe repetitious crystal structure. Butinformation about the molecules thatform a crystal is also contained in thediffraction pattern, encoded in the in-tensities and phases of the diffractionspots. Working backward from thisinformation to a three-dimensional mo-lecular structure is simple: The rulesays that each spot is a term in theFourier transform of the molecularstructure. Unfortunately, reality israrely as kind as theory.

The trouble is that photographic filmrecords intensity, but is insensitive tophase. Crystallographers have invent-ed a host of ingenious dodges to getaround the phase problem (PHYSICSTODAY, November 1982, page 17). Onestandard method is to substitute heavyatoms selectively at various positionsin the molecule being studied. Becauseheavy atoms reflect x rays more strong-ly than light ones, they act as markers.By comparing diffraction patterns ofthe modified and unmodified mole-cules, one can often infer enough infor-mation to solve the structure. Forsmaller structures the heavy atomgives enough information that one cansolve the structure completely, but it

KARLE

may also severely distort the structure.To verify that the structure is correctthe crystallographers must first makesure it makes good chemical sense(proper bond distances and angles, andgood packing distances) and that itreproduces the measured diffractionpattern. Most of the triumphs of x-raycrystallography before Hauptman andKarle's work were based on substitu-tion techniques and inspired guess-work. A typical thesis project in thosebad old days was the determination ofthe structure of one or two mid-sizedmolecules.

Bridging. Just after the Second WorldWar Karle and his wife Isabella, nowworking at the Naval Research Labora-tory, became interested in developingthe quantitative aspects of electrondiffraction of gas molecules. In a gasthe molecules are oriented randomly,so the resulting diffraction patterns arenot spots, but diffuse rings. The Karlesfound that the analysis was simplifiedif they placed physical restraints on theproblem, such as requiring that theFourier transform be nonnegative, be-cause it represents a probability den-sity. "This nonnegativity worked sowell, it was so exciting," recalls Karle,"that I began to think of other applica-tions."

Around this time, in 1947, Haupt-man came to the Naval Research Labo-ratory and joined the Karles. "The warwas over, I got out of the Navy andabout all I knew was that I wanted to

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